Scatter Diagrams and Correlation
4.1 & 4.3
2 Variables ●In many studies, we measure more than one variable for each individual
●Some examples are
Rainfall amounts and plant growth
Exercise and cholesterol levels for a group of people
Height and weight for a group of people
●In these cases, we are interested in whether the two variables have some kind of a relationship
2 Variables ●When we have two variables, they could be related in one of several different ways
They could be unrelated
One variable (the explanatory or predictor variable) could be used to explain the other (the response or dependent variable)
One variable could be thought of as causing the other variable to change
●In this chapter, we examine the second case … explanatory and response variables
Lurking Variable •Sometimes it is not clear which variable is the explanatory variable and which is the response variable
•Sometimes the two variables are related without either one being an explanatory variable
•Sometimes the two variables are both affected by a third variable, a lurking variable, that had not been included in the study
Example of a Lurking Variable ●A researcher studies a group of elementary school children
Y = the student’s height
X = the student’s shoe size
●It is not reasonable to claim that shoe size causes height to change
●The lurking variable of age affects both of these two variables
More Examples ●Rainfall amounts and plant growth
Explanatory variable – rainfall
Response variable – plant growth
Possible lurking variable – amount of sunlight
●Exercise and cholesterol levels
Explanatory variable – amount of exercise
Response variable – cholesterol level
Possible lurking variable – diet
Scatter Diagram •The most useful graph to show the relationship between two quantitative variables is the scatter diagram
•Each individual is represented by a point in the diagram
•The explanatory (X) variable is plotted on the horizontal scale
•The response (Y) variable is plotted on the vertical scale
Scatter Diagram •An example of a scatter diagram
•Note the truncated vertical scale!
Relations ●There are several different types of relations between two variables
A relationship is linear when, plotted on a scatter diagram, the points follow the general pattern of a line
A relationship is nonlinear when, plotted on a scatter diagram, the points follow a general pattern, but it is not a line
A relationship has no correlation when, plotted on a scatter diagram, the points do not show any pattern
Positive vs. Negative •Linear relations have points that cluster around a line
•Linear relations can be either positive (the points slants upwards to the right) or negative (the points slant downwards to the right)
Positive ●For positive (linear) associations
Above average values of one variable are associated with above average values of the other (above/above, the points trend right and upwards)
Below average values of one variable are associated with below average values of the other (below/below, the points trend left and downwards)
●Examples
“Age” and “Height” for children
“Temperature” and “Sales of ice cream”
Negative ●For negative (linear) associations
Above average values of one variable are associated with below average values of the other (above/below, the points trend right and downwards)
Below average values of one variable are associated with above average values of the other (below/above, the points trend left and upwards)
●Examples
“Age” and “Time required to run 50 meters” for children
“Temperature” and “Sales of hot chocolate”
Nonlinear •Nonlinear relations have points that have a trend, but not around a line
•The trend has some bend in it
Not Related •When two variables are not related
•There is no linear trend
•There is no nonlinear trend
•Changes in values for one variable do not seem to have any relation with changes in the other
Examples ●Examples of nonlinear relations
“Age” and “Height” for people (including both children and adults)
“Temperature” and “Comfort level” for people
●Examples of no relations
“Temperature” and “Closing price of the Dow Jones Industrials Index” (probably)
“Age” and “Last digit of telephone number” for adults
Linear Correlation Coefficient •The linear correlation coefficient is a measure of the
strength of linear relation between two quantitative variables
•The sample correlation coefficient “r” is
•This should be computed with software (and not by hand) whenever possible
Linear Correlation Coefficient ●Some properties of the linear correlation coefficient
r is a unitless measure (so that r would be the same for a data set whether x and y are measured in feet, inches, meters, or fathoms)
r is always between –1 and +1
Positive values of r correspond to positive relations
Negative values of r correspond to negative relations
Linear Correlation Coefficient ●Some more properties of the linear correlation coefficient
The closer r is to +1, the stronger the positive relation … when r = +1, there is a perfect positive relation
The closer r is to –1, the stronger the negative relation … when r = –1, there is a perfect negative relation
The closer r is to 0, the less of a linear relation (either positive or negative)
Examples ●Examples of positive correlation
●In general, if the correlation is visible to the eye, then it is likely to be strong
•Examples of positive correlation
Strong Positive r = .8
Moderate Positive r = .5
Very Weak r = .1
Negative ●Examples of negative correlation
●In general, if the correlation is visible to the eye, then it is likely to be strong
Strong Negative r = –.8
Moderate Negative r = –.5
Very Weak r = –.1
●Examples of negative correlation
Nonlinear ●Nonlinear correlation
●Has an r = 0.1, but the difference is that the nonlinear relation shows a clear pattern (or lack of)
Correlation… ●Correlation is not causation!
●Just because two variables are correlated does not mean that one causes the other to change
●There is a strong correlation between shoe sizes and vocabulary sizes for grade school children
Clearly larger shoe sizes do not cause larger vocabularies
Clearly larger vocabularies do not cause larger shoe sizes
●Often lurking variables result in confounding
Coefficient of Determination •R2 – coefficient of determination, measures the proportion of total
variation in the response variable that is explained by the least-squares
regression line.
Example •Weight of Car Vs. Miles Per Gallon
•Y = -.007036x + 44.8793
•R = -964086
•R2 = 929461
93% of the variability in miles per gallon can be explained by its linear relationship with the weight.
7% of miles per gallon would be explained by other factors
Calculators •Draw a scatter diagram
Age HDL Cholesterol 38 57
42 54
46 34
32 56
55 35
52 40
61 42
61 38
26 47
38 44
66 62
30 53
51 36
27 45
52 38
49 55
39 28
AGE VS. HDL CHOLESTEROL A doctor wanted to determine
whether a relation exists between a male’s age and his HDL (so-called
good) cholesterol. He randomly selected 17 of his patients and
determined their HDL cholesterol levels. He obtained the following
data.
• New Document
• Insert Lists & Spreadsheet
• Column A (age) Column B (HDL)
• Type in Data
• Insert Data & Statistics (Ctrl I)
• Put “age” on x-axis (explanatory)
• Put “HDL” on y-axis (response)
• Observe Data (does there appear to be a relationship)
• Menu
• 6:regression
• Linear Regression
Insert Calculator Page (Ctrl I) Run Linear Regression
Menu 6: Statistics 1: Stat Calculations 3: Linear Regression X List “age” Y List “HDL” ENTER Record equation, r-value, and r2 - value
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